## Limit Operators and Their Applications in Operator TheoryThis text has two goals. It describes a topic: band and band-dominated operators and their Fredholm theory, and it introduces a method to study this topic: limit operators. Band-dominated operators. Let H = [2(Z) be the Hilbert space of all squared summable functions x : Z -+ Xi provided with the norm 2 2 X IIxl1 :=L I iI . iEZ It is often convenient to think of the elements x of [2(Z) as two-sided infinite sequences (Xi)iEZ. The standard basis of [2(Z) is the family of sequences (ei)iEZ where ei = (. . . ,0,0, 1,0,0, . . . ) with the 1 standing at the ith place. Every bounded linear operator A on H can be described by a two-sided infinite matrix (aij)i,jEZ with respect to this basis, where aij = (Aej, ei)' The band operators on H are just the operators with a matrix representation of finite band-width, i. e. , the operators for which aij = 0 whenever Ii - jl > k for some k. Operators which are in the norm closure ofthe algebra of all band operators are called band-dominated. Needless to say that band and band dominated operators appear in numerous branches of mathematics. Archetypal examples come from discretizations of partial differential operators. It is easy to check that every band operator can be uniquely written as a finite sum L dkVk where the d are multiplication operators (i. e. |

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### Contents

Fredholmness of Banddominated Operators | 30 |

4 | 76 |

converges strongly as n oo The strong limit of this sequence is called the limit | 136 |

Convolution Type Operators on | 153 |

Pseudodifferential Operators | 201 |

XDr | 205 |

Pseudodifference Operators | 267 |

Finite Sections of Banddominated Operators | 303 |

Axiomatization of the Limit Operators Approach | 345 |

Bibliography 375 | 374 |

### Other editions - View all

Limit Operators and Their Applications in Operator Theory Vladimir Rabinovich,Steffen Roch,Bernd Silbermann No preview available - 2004 |

Limit Operators and Their Applications in Operator Theory Vladimir Rabinovitch,Steffen Roch,Bernd Silbermann No preview available - 2004 |

### Common terms and phrases

A(BUC(R aſa assertion Banach algebra Banach space band operators band-dominated operators belongs C*-algebra C*-subalgebra characteristic function choose closed subalgebra compact operators compression continuous function Corollary coset defined Definition discrete equivalent essential spectrum exists Fredholm operator Further g of h Gelfand transform h e H half-space Hence Hilbert space homomorphism identity operator implies integral Lemma Let a e let h limit operator Ah lº(Z locally invertible LP(R m e S^T mapping maximal ideal multiplication operator norm operator of multiplication operator Op(a operator spectrum P-compact P-Fredholm P-strong proof of Theorem Proposition pseudodifference operators pseudodifferential operators respect Schrödinger operators sequence h e slowly oscillating functions SO(R strong convergence strong limit operator subsequence g tends to infinity topology uniformly bounded uniformly invertible unitary operator whence Wiener algebra zero